The following article is published:
Scott Johnson and Hans Walser: The Pop-Up Octahedron. Mathematics in School. Vol. 25, No. 5, November 1997, 2-4
Back in 1988 we published a set of Maths Resource pages on p0p-up polyhedra (Cassell, 1988) which were well received at the time. The 'toys' developed by the two authors as described here take the art somewhat further. Of particular interest is the fact that the principles employed can be applied to many other polyhedra and this article complements another, more theoretical, article planned to be published concurrently in our sister publication, the Gazette (]ohnson & Walser, 1997)
by Scott Johnson and Hans Walser
In this article we discuss how to build a mathematical toy. First we describe how to construct a pop-up octahedron. This octahedron may be pressed flat into a planar configuration and then when you either let go, or toss it into the air, it pops (making a noise) into the shape of the regular octahedron. Figure l (a) illustrates the entire spatial octahedron. The Figures 1 (b) and 1 (c) depict the intermediate stages as you press it flat, by exerting downward pressure on the top vertex (the North Pole) while holding the bottom vertex (the South Pole) in a fixed position. The dark portions of Figures 1 (b) and 1 (c) show the inner cavity of the octahedron which will be visible between the two edges that separate as the model goes down into a planar shape (and, in fact, on your finished model you will then be able to see the rubber band that makes the model pop-up). Figure 1 (d) shows how the model looks in its flattened position. In terms of our global description the model opens along a longitudinal line, connecting the North and South Poles, running along edges of the octahedron. The outer hexagonal part of the border in the flattened position is part of the original equator.
Fig. 1 Pressing down
The flattened situation consists of two layers with four equilateral triangles in each layer. Observe that, since every angle of an equilateral triangle is 60ˇ, you have at the centre (where the two former poles now meet) a gap of 120ˇ. If we were to fill this gap with two additional equilateral triangles on the top layer and two directly beneath those on the bottom layer we would form two regular hexagons - and that model would not be deformable into a polyhedron having these 12 equilateral triangles as faces. Such a model would either remain flat, or would flex, along the six edges radiating from the center, in space. However, if you added only one equilateral triangle on the top layer and one directly beneath it on the bottom layer you can construct another pop-up polyhedron, which is known as the pentagonal dipyramid shown in Figure 2(a). Although this 10-faced polyhedron is not as regular as the octahedron (not all vertices are contiguous with the same number of edges) it does have some interesting properties and uses (it may serve, for example, as a random number generator for the ten digits 0 through 9).
If, instead of adding a triangle to each layer, you remove one triangle from each layer (one lying above the other) you will be able to construct a 6-faced polyhedron known as the triangular dipyramid shown in Figure 2(b). Again, this 6-faced polyhedron is less regular than the octahedron.
Fig. 2 Double pyramids
What happens if you remove two contiguous equilateral triangles from each layer (where they both lie above each) from the configuration in Figure 1 (d)? You will have, in fact, four equilateral triangles. Why don't you get the regular tetrahedron shown in Figure 3? If you don't see the answer to this construct the configuration and examine it closely.
Fig. 3 The regular tetrahedron
What we describe is how we built our models. However, you may find other materials that work just as well. You should experiment and find out what works best for you.
We built our models from foamboard (this is a 5 mm thick piece of foam covered with paper on both sides), and sometimes from cardboard, or thin plywood (about 3 mm thick). First you draw the required eight equilateral triangles on the foamboard. Of course the size of these triangles is up to you, but all of them need to be congruent. We built our models with an edge length of 10 cm. To cut the foamboard we used a sharp knife - a Swiss army knife will work, and so will a surgeon's scalpel (these were the knives the two authors used).
At every edge of the octahedron - except at the two edges that separate along the longitudinal line running from the North to the South Pole - we have to connect the triangles by hinges. Figure 4 shows how we made the hinges. We first taped the two parts on the inner side with pieces of fiberglass reinforced tape as shown in Figure 4(a), then we closed the hinges and taped the back at the outside of the two parts as shown in Figure 4(b).
Fig. 4 The construction of a hinge
These hinges will allow motions of the two joined faces between 0ˇ and 180ˇ as shown in Figure 5. It is impossible (and unnecessary here) to make the angle between the joined faces more than 180ˇ.
Fig. 5 Range of a hinge
Now a word of caution. You cannot tape all the connected edges by hinges in the manner depicted in Figure 4, because it is impossible to open and close every edge of the finished model in an arbitrary way. Therefore think before taping. Our procedure (you may find a better one) is to first tape each layer separately - this is no problem. Then we tape one of the four equatorial hinges connecting the two layers. The taping of the remaining three hinges of the equator is more sophisticated, since these are the hinges that now won't open in the manner of Figure 4(a). But with a little forethought and ingenuity you should be able to manage it - and remember no man, or woman, is born a master.
At this point you have all the geometrical parts of the octahedron, but the two layers are like a balloon without air. You have to supply the air - and this is done by connecting a rubber band to two of the already taped edges - and this animates the model.
What is required is to attach the rubber band at two more or less opposite points R1 and R2 of the outer borderline of edges shown on the flat two-layered model as shown in Figure 6.
Fig. 6 Points of attachment of the rubber band
To attach the rubber band at these two points first reinforce the outside hinge along the edges containing and with one or two additional pieces of tape as shown in Figure 7 (a).
Fig. 7 The attachment of the rubber band
Next punch a little hole and pull the rubber band through it (a deformed paper clip may be used like a crochet hook to do this) as shown in Figure 7 (b). Then lock the rubber band in place by a small bolt, made of metal (cut a paper clip) or wood (part of a toothpick) trough it along the edge as shown in Figure 7(c). Then to affix the bolt cover it with another piece of tape. Finally, attach the other end of the rubber band (that is, the part of the robber band diametrically opposite the part of the rubber band already attached) to the other edge. This will involve stretching the rubber band. It is sometimes useful to pull the rubber band through the hole by the help of a piece of thread and a needle. You may need to experiment with rubber bands to get the one of just the right length. However, after you have made one model you will soon get a sense of how long, or short, you need the rubber band to be.
When your model is complete either press it flat onto a table and lift your hand quickly, or press it flat and toss it into the air. It should pop-up in either case and make a snapping sound.
The pop-up octahedron is only one example of a lot of other possibilities of pop-up polyhedra. We have written about some of them in the Gazette Johnson & Walser, 1997) and Hilton and Pedersen have made pop-up models with one piece of cardboard as described in their book (Hilton & Pedersen, 1994). However, we wish to emphasize that the actual construction of any model is an art and you may very well discover other ways and other material with which to build pop-up models of cubes, tetrahedra, as well as other polyhedra. We would be delighted to hear of any improvement on our method of construction and about any new models you invent. What you really must do is make these models yourself to appreciate both their intrinsic beauty and their underlying structure.
The authors would like to thank Jean Pedersen for her helpful suggestions during the preparation of this manuscript.
Hilton, P. and Pedersen, J. (1994) Build Your Own Polyhedra, AddisonWesley.
Johnson, S. and Walser, H. (1997) Pop-Up Polyhedra, The Mathematical Gazette, November, 1997 (to appear).
Cassell, D. (1988) Pop-Up Polyhedra, M. Res 77-79 Mathematics in School, 17,1
Scott Johnson (Student), Department of Mathematics, Santa Clara University, Santa Clara, CA 95053, USA
Dr Hans Walser, Department of Mathematics, Santa Clara University, Santa Clara, CA 95053, USA
Mathematisches Institut, Basel University, Rheinsprung 21, CH-4051 Basel